I don't think there's much written on this idea. It's used in statistical physics when you want your argument to be independent of having a discrete collection of matter, so integrals can replace clumsy sums and stuff like that. You can only apply it when doing that is a physical reality. For instance, in systems where adding more particles would change the charge distribution, it wouldn't leave your problem unchanged if you added more particles.

For example if you were trying to figure out the probability of a transcription factor binding to a certain place on a genome, it would be okay, in your reasoning, to assume that the number of binding sites was very large and that as the genome got infinitely long so did the number of TF binding sites. However, if these binding sites could maybe attract each other and form bonds, this would obviously change the whole problem. The network of connections made by the binding sites with each other may not scale with adding more sites.

So when [tex]N\rightarrow \infty[/tex], [tex]\varphi_{H}\rightarrow 0[/tex].

So I can say microcanonical and canonical enseble are equivalent. But can you give me I don't know three examples when [tex]\left\langle H \right\rangle \propto N[/tex] is not correct. Because from this I can say that if and only if [tex]\left\langle H \right\rangle \propto N[/tex] transition from statistical physics to thermodynamics is possible?